5.1.17. (-) Use Lemma 5.1.18 to prove that x (G) ≤ 4 for the graph G below.
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- 4. Let G=(V, E) be the following undirected graph: V = {1, 2, 3, 4, 5, 6, 7, 8} E = {(1, 7) (1, 4), (1, 6), (6, 4), (7, 5), (2, 5), (2, 4), (4, 3), (4, 8), (8, 3), (3, 5)}. a. Draw G. b. Is G connected? c. Give the adjacency matrix for the graph G given above. d. Determine whether the graph has a Hamiltonian cycle. If the graph does have a Hamiltonian cycle, specify the nodes of one. If it does not, prove that it does not have one. e. Determine whether the graph has a Euler cycle. If the graph does have a Euler cycle, specify the nodes of one. If it does not, prove that it does not have one.2. A graph G = (V, E) satisfies |E| < 3 × |V| – 6. The min-degree of G is defined as min,ev{degree(v)}. The min-degree of G cannot be1.6 If G is a simple graph of with number of vertices n > 0, show that deg(v) id(v) = od(v) = q. VEV VEV
- Question 2: Let G be a planar graph on at least 4 vertices. Suppose that the minimum degree of G is 5. Show that the number of vertices of degree 5 in G is at least 12.2. Prove that in a graph G, Sc V(G) is an independent set if and only if S is a vertex cover and hence a(G)+B(G) = n(G).Question 1 Let S be a subset of vertices in G, and let C be the complement graph of G (where uv is an edge in C if and only if uv is not an edge in G).Prove that for any subset of vertices S, S is a vertex cover in G if and only if V\S is a clique in C.Note: this is an if and only if proof, i.e. you need to show both directions for full credit. Question 2 Part 4.1 implies the following result (which you may use without proof): G has a vertex cover of size at most k if and only if the complement of G has a clique of size at least n−k.Use this fact to give a reduction from VertexCover to Clique. Your solution should have the following two steps:i) First, show the reduction: specify how the inputs to VertexCover, G and k, can be transformed to a valid input pair, H and l, for Clique. Make sure to explain why this takes polynomial time.ii) Second, show that the answer to Clique(H,l) can be converted to the answer of VertexCover(G,k). One possibility is to explain how a YES answer to…
- 1. Let G be a graph with spectrum ( ). 3,3,6 (a) How many vertices does G have? Is G bipartite? (b) What is the minimum polynomial of G? (c) What is the best bound you can find for A(G)? (d) What is the best bound you can find for the size of G? (e) What is the best bound you can find for the diameter of G? (f) What is the best bou you can find for the chromatic number of G?Problem 3. Write down a topological ordering for the directed graph if Figure 1 (given as a sequence of node names (e.g., "c,a,e,f,...")), or prove that no topological ordering is possible.1. Let G be a tree and let L be the set of leaves in G (the vertices of degree 1). (a) If G contains a vertex of degree k, show that |L| > k. (b) Let f be a graph isomorphism from G to G. Prove that f(L) = L. (c) Prove that either there is a vertex v e V(G) such that f(v) = v or there is an edge {x, y} € E(G) such that {f(x), f(y)} = {x, y}. (Hint: Induction on |V(G)|; in the induction step consider a restriction of ƒ to a subset of vertices.)
- Please solve and show all work. 21.1-2 Professor Sabatier conjectures the following converse of Theorem 21.1. Let G = (V, E) be a connected, undirected graph with a real-valued weight function w defined on E. Let A be a subset of E that is included in some minimum spanning tree for G, let (S, V – S) be any cut of G that respects A, and let (u, v) be a safe edge for A crossing (S, V – S). Then, (u, v) is a light edge for the cut. Show that the professor’s conjecture is incorrect by giving a counterexample.For the graph G of Figure 2.11, construct a 3-regular graph H containing G as an induced subgraph (a) using the proof of Theorem 2.7. What is the order of H? (b) such that H has the smallest possible order. What is this order? G: Figure 2.11: The graph G in Exercise 2.22Question 1: In graph theory, a graph X is a "complement" of a graph F if which of the following is true? Select one: a. If X is isomorph to F, then X is a complement of F. b. If X has half of the vertices of F (or if F has half of the vertices of X) then X is a complement of F. c. If X has the same vertex set as F, and as its edges ONLY all possible edges NOT contained in F, then X is a complement of F. d. If X is NOT isomorph to F, then X is a complement of F. Question 2: Which statement is NOT true about Merge Sort Algorithm: Select one: a. Merge Sort time complexity for worst case scenarios is: O(n log n) b. Merge Sort is a quadratic sorting algorithm c. Merge Sort key disadvantage is space overhead as compared to Bubble Sort, Selection Sort and Insertion Sort. d. Merge Sort adopts recursive approach